Fermion Monte Carlo without fixed nodes: A Game of Life, death and annihilation inSlater Determinant space

George H. Booth(a), Alex J. W. Thom(a,b), and Ali Alavi(a)∗(a)University of Cambridge, Chemistry Department,Lensfield Road, Cambridge CB2 1EW, U.K. and

(b)University of California Berkeley, Department of Chemistry, Berkeley, CA 94720 U.S.A.(Dated: July 12, 2009)

We have developed a new Quantum Monte Carlo method for the simulation of correlated many-electron systems in Full Configuration-Interaction (Slater Determinant) spaces. The new method isa population dynamics of a set of walkers, and is designed to simulate the underlying imaginary-timeSchrodinger equation of the interacting Hamiltonian. The walkers (which carry a positive or negativesign) inhabit Slater determinant space, and evolve according to a simple set of rules which includespawning, death and annihilation processes. We show that this method is capable of convergingonto the Full Configuration-Interaction (FCI) energy and wavefunction of the problem, withoutany a priori information regarding the nodal structure of the wavefunction being provided. Walkerannihilation is shown to play a key role. The pattern of walker growth exhibits a characteristicplateau once a critical (system-dependent) number of walkers has been reached. At this point, thecorrelation energy can be measured using two independent methods – a projection formula anda energy shift; agreement between these provides a strong measure of confidence in the accuracyof the computed correlation energies. We have verified the method by performing calculations onsystems for which FCI calculations already exist. In addition, we report on a number of new systems,including CO, O2, CH4 and NaH – with FCI spaces ranging from 109 to 1014, whose FCI energieswe compute using modest computational resources.

PACS numbers:

INTRODUCTION

It has long been known that the only major obsta-cle preventing the exact numerical simulation of many-electron systems via stochastic methods such as Diffusionquantum Monte Carlo (DMC) [1] or the related Green’sfunction Monte Carlo (GFMC) [2] is the Fermion signproblem [3]. This problem stems from the antisymme-try property of many-electron wavefunctions to electronexchange, which leads to wavefunctions which have bothpositive and negative amplitudes. Since the Schrodingerequation can be viewed as a diffusion equation in imag-inary time, its lowest energy solution is in general node-less and symmetric, and therefore does not satisfy therequired Fermion antisymmetry. Stochastic propagationleads exponentially quickly to this undesired solution.One way to prevent this “Boson catastrophe” is to con-strain the propagation to disjoint areas of similar signusing the fixed-node approximation [4–7], a procedurewhich would be exact if the applied nodal boundaries co-incided with the exact nodal hypersurface of the ground-state electronic wavefunction. However, in practice, thisis not the case, and it has proven extremely difficult toimprove the fixed-node surface towards the exact one.The Fermion sign problem is thus recognised as one ofthe most important unsolved problems of computational

∗Electronic address: asa10@cam.ac.uk

theoretical physics and chemistry. A key question iswhether the exact nodal hypersurface can emerge dur-ing the course of a simulation. Such a simulation wouldtherefore not require any a priori information regardingthe nodes of the exact wavefunction. In this paper wedescribe a new quantum Monte Carlo (QMC) method inwhich this highly desirable property is shown to arise,for systems described with basis-sets commonly used inquantum chemistry. Our method is shown to convergeon to the Full Configuration-Interaction (FCI) solution,i.e. the exact wavefunction and energy for the basis-setunder consideration [8–11]. In addition to reproducingexisting FCI calculations, we have used this method topredict the FCI energies of several molecules which havenot been reported to date. We also report a benchmarkstudy of the Ne atom (correlating all electrons) in sev-eral basis sets up to cc-pCVQZ, which has a FCI basisexceeding 1014 determinants. That is over 10,000 timeslarger than the largest FCI calculation reported to date[12]. Our method unifies QMC and FCI in a way whichprofoundly extends the scope of both techniques.

There are three ingredients to the new method, whichtake elements from DMC and FCI: (i) In commonwith DMC, we perform a long-time integration of theimaginary-time Schrodinger equation; however, in con-trast to DMC, this is achieved in a space of Slater de-terminants. In addition, the propagation step in our al-gorithm differs from DMC in that it consists purely ofpopulation dynamics (i.e. walker birth and death pro-cesses). There are no diffusive moves as such. (ii) In

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common with DMC, the instantaneous wavefunction isrepresented using “walkers”, rather than amplitude co-efficients, the latter being the case in FCI. The repre-sentation using “walkers” enables us to describe stochas-tically the FCI wavefunction without storing all ampli-tudes simultaneously. (iii) Each walker carries a positiveor negative sign. The key ingredient in our algorithmis walker annihilation, in which pairs of walkers of op-posite sign which coincide on the same determinant areremoved from the simulation. We show that an algo-rithm based on these three ideas converges on the ex-act solution to the Hamiltonian expressed in the basis,which owing to the Fermion antisymmetry of the Slaterdeterminant basis, will be the lowest Fermionic solutionavailable in this basis. It should be noted that work-ing in Slater determinant space does not circumvent thesign problem, even though it prevents convergence to aBosonic solution. This is because the FCI wavefunctiondoes not (in general) have strictly non-negative ampli-tudes in this space. We note that walker annihilationhas been previously suggested in the context of nodalrelease GFMC and DMC [13–16]. However, as noted inthese studies, only low-dimensional phase-spaces could betreated owing to the formidable difficulties in achievingcancellation in continuum spaces. Instead, by working indiscrete Slater determinant spaces, we find that walkerannihilation proves an essential and effective componentof a Fermion Monte Carlo method.

There have been several proposals to simulateFermionic systems in determinantal spaces usingauxilliary-field Monte Carlo(AFMC) [17–22]. However,in their exact formulation, auxillary-field methods suf-fer from exponentially large statistical noise in the limitof large imaginary-time propagation, which is necessaryto project out the ground state. In order to stabilisethis problem, a phaseless-approximation AFMC methodhas been proposed [21], with promising energies (i.e. towithin a few milli Hartrees of known FCI energies) whenapplied to molecular systems [22]. However, like thefixed-node approximation, the phaseless approximationis an uncontrolled approximation which may be difficultto improve upon.

Below we give motivation and elaborate on our algo-rithm, before showing applications of the method to sev-eral real physical systems.

MOTIVATION AND DERIVATION OF THEALGORITHM

In the FCI method, we seek a wavefunction Ψ0

which satisfies the time-independent Schrodinger equa-tion, HΨ0 = E0Ψ0, as a variationally optimised linearcombination of Slater determinants {|Di〉}. These areantisymmetric functions in which N orbitals are chosenout of 2M spin-orbitals {φ1, φ2, ..., φ2M} and constructed

as follows:

|Di〉 ≡ |Dn1,n2,..,nN〉 = a†n1

a†n2....a†nN

| 〉 (1)

=1√N !|φn1φn2 .....φnN

|, n1 < n2 < ... < nN(2)

a†i is a Fermion creation operator for spin-orbital φi. Inthis work, the orbitals used are real, canonical Hartree-Fock orbitals. The size of the Slater determinant space(NFCI) is on the order of

(M

N/2

)2for a spin-unpolarised

system, a number that grows factorially with M and N(although symmetry restrictions for some systems canreduce this by up to an order of magnitude). The FCIwavefunction is expressed as:

ΨFCI0 =

∑

i

Ci|Di〉 (3)

where the CI coefficients {Ci} satisfy an eigenvector prob-lem:

∑

j

〈Di|H|Dj〉Cj = EFCI0 Ci. (4)

EFCI0 is the lowest energy solution available in this ba-

sis, an upper bound to the exact energy. Owing to thefact that the off-diagonal Hamiltonian matrix elementsare not all of the same sign, the CI coefficients can bepositive or negative. The “sign” structure of the FCIwavefunction is given by a vector whose components aresign(Ci) or 0 if Ci = 0. A trial wavefunction has thecorrect sign structure only if the sign of every compo-nent matches those of this vector (up to an overall sign,since −ΨFCI

0 is an equally valid solution with the sameenergy). In practice, it turns out to be impossible topredict sign(Ci) without a knowledge of Ci, and this, inessence, is the manifestation of the Fermion sign problemin the discrete Slater determinant space. In FCI, the CIcoefficients are obtained via a (non-stochastic) iterativediagonalisation method. Whilst it is recognised that theFCI method is the most robust method to treat electroncorrelation, its scope is greatly limited by the prohibitivecomputational requirements (especially storage) of suchiterative diagonalisation in the full space of Slater deter-minants. For example the Davidson [23] method or thepreconditioned conjugate-gradient method [24] requiresat least two vectors of length equal to the FCI space to bestored, and often many more. The largest molecular FCIcalculation to date is the N2 molecule [12], and has≈ 1010

determinants in D2h symmetry. As Nicholas Handy hasobserved [25], “unless something unexpected happens, itis unlikely that size will be much exceeded”. Inspired bythe DMC method, rather than attempt a direct diagonal-isation of the FCI Hamiltonian, we propose to simulatethe imaginary-time Schrodinger equation by performinga stochastic population dynamics on an evolving set of“walkers” which live and propagate in Slater determinant

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space. However, we do not impose any prior knowledgeon the signs of the Slater determinants.

The imaginary-time Schrodinger equation provides thefundamental starting point of all “projector” techniques.It states that the imaginary-time derivative of any wave-function Ψ is given simply by the Hamiltonian acting on−Ψ, namely:

∂Ψ∂τ

= −HΨ (5)

Given a starting wavefunction Ψ(τ = 0), the wavefunc-tion Ψ(τ) for arbitrary τ is proportional to:

Ψ(τ) ∝ e−τHΨ(τ = 0) (6)

In order to project onto the Fermionic ground-state,Ψ(τ = 0) must chosen to be a fully-antisymmetric func-tion with an overlap with the ground state. Usually asingle determinant function with the correct symmetry,such as the Hartree-Fock determinant D0, suffices. Thelong-time limit of Eq. (6) then projects out the com-ponents of D0 on excited-states, leaving the Fermionicground-state, Ψ0:

Ψ0 = limτ→∞

e−τ(H−E0)D0 (7)

where the constant of proportionality e+τE0 has been in-troduced to keep the ground-state contribution from de-caying to zero. The aspiration of all Monte Carlo pro-jector methods is to realise this long-time limit by per-forming a stochastic integration of Eq. (5). In normalDiffusion Monte Carlo, however, the integration is notperformed on an antisymmetrised space, and thereforeadmits solutions with Bosonic character, which rapidlygrow to dominate unless constrained in some manner,e.g. by the fixed-node approximation. An integration ina pure antisymmetric subspace would not suffer from thiseffect, and this is may be a help in reducing the sign prob-lem for Fermion systems [26, 27]. Our first aim, therefore,is to develop an analogue to Eq. (5) in a Slater determi-nant basis, in a form which can be integrated stochasti-cally.

To this end, let us define a matrix K, whose elementsare the matrix elements between Slater determinants ofthe Hamiltonian, with the Hartree-Fock energy EHF sub-tracted from the diagonal elements:

Kij ≡ 〈Di|K|Dj〉 = 〈Di|H|Dj〉 − EHFδij (8)

The diagonal matrix-elements of K are therefore all pos-itive (or zero), whilst the off-diagonal elements are sim-ply the off-diagonal matrix elements of the Hamiltonian.With this definition, the lowest energy eigenvalue of K isEFCI

0 −EHF, which is the correlation energy, Ecorr for theproblem. The matrix elements of H can be computed interms of the one-electron and two-electron integrals us-ing standard methods, which are reviewed in AppendixA.

Writing

Ψ(τ) =∑

i

Ci(τ)|Di〉 (9)

and substituting into Eq. (5), we obtain a set of coupledlinear first-order differential equations for the CI coeffi-cients in terms of the K matrix:

−dCi

dτ=

∑

j

(Kij − Sδij)Cj (10)

= (Kii − S)Ci +∑

j 6=i

KijCj (11)

where an arbitrary “energy shift”, S, has been introducedinto the diagonal terms, whose role will be populationcontrol, to be discussed later. It is evident that if we in-stantaneously have a vector C of amplitudes whose com-ponents Ci satisfy

∑

j

KijCj = SCi (12)

then dC/dτ = 0, making this vector stationary. In addi-tion, by virtue of Eq. (12), this vector is an eigenstate ofthe K matrix (and hence of the H), with eigenvalue S. Itfollows that if S equals the correlation energy Ecorr, thenthe stationary state is the ground state of the H matrix.Furthermore, if we start from an arbitrary set of Ci am-plitudes (which does not satisfy Eq. (12)) and integratethe above set of coupled differential equations, the long-time solution leads to the ground-state eigenvector. Thiscan be proven by decomposing the starting vector intothe eigenstates of H, and noting that the components onthe excited states would decay exponentially with time,leaving the ground-state eigenvector.

A direct numerical integration of Eq. (11) requires thefull set of Ci coefficients to be available at each time step,which is prohibitive. Instead, in the spirit of DMC, letus consider a population of Nw walkers; each walker αis located on a determinant iα, and has a sign sα = ±1.We now define the Ci amplitude on determinant |Di〉 tobe proportional to the signed sum of walkers (Ni) ,

Ci ∝ Ni =∑α

sαδi,iα (13)

(δi,iα is the discrete Kronecker delta, and equals one ifiα = i, and is otherwise zero). According to Eq. (13), Ni

can be positive or negative. However, the total numberof walkers Nw is defined to be the sum over the absolutevalues of the Ni:

Nw =∑

i

|Ni| (14)

and is always positive.

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A POPULATION DYNAMICS ALGORITHM

We introduce a population dynamics algorithm whichsimulates the set of coupled differential equations in Eq.(11). The algorithm consists of three steps performed ateach timestep whose length is δτ :

(i) The spawning step: for each walker α (located onDiα), we select a (coupled) determinant Dj withnormalised probability pgen(j|iα) and attempt tospawn a child there with probability

ps(j|iα) =δτ |Kiαj|pgen(j|iα)

(15)

If a spawning event is successful, (i.e. if ps ex-ceeds a uniformly chosen random number between0 and 1), then the sign of the child is determinedby the sign of Kiαj and the sign of the parent: itis the same sign as the parent if Kiαj < 0, andopposite to the parent otherwise. Our method tocompute the generation probabilities pgen is givenin Appendix B. If ps > 1, then multiple copies ofwalkers are spawned on j (namely with probability1, bpsc walkers are spawned, and with probabilityps − bpsc an additional walker is spawned).

In general, the number of newly-spawned walkers(Ns) is much smaller than number of parent walk-ers (Nw), since the time step δτ is such that theprobability to spawn is quite low. Typically wefind Ns ∼ 10−4Nw.

(ii) The diagonal death/cloning step: for each (parent)walker compute

pd(iα) = δτ(Kiαiα − S) (16)

If pd > 0, the walker dies with probability pd,and if pd < 0 the walker is cloned with proba-bility |pd|. The death event happens immediately,and such a parent does not participate in the fol-lowing (annihilation) step to be described shortly.Cloning events are quite rare, and only occur forS > 0, and even then only on determinants forwhich 〈Di|K|Di〉 < S. In simulations where we de-sire to grow the number of walkers rapidly, a pos-itive value of S is adopted, and this can lead tocloning events. However, more often, the value ofS is negative (as it tries to match the correlationenergy), and in such cases there can be no cloningevents at all.

Both the spawning step and the diagonal deathstep can be done without reference to other walk-ers. Therefore, these two steps are “embarrassinglyparallel”, and can be performed without communi-cation overhead on a parallel machine.

(iii) The annihilation step: In this (final) part of thealgorithm, we run over all (newly-spawned, clonedand surviving parent) walkers, and annihilate pairsof walkers of opposite sign which are found to be onthe same determinant. Each time an annihilationevent occurs, the corresponding pair is removedfrom the list of walkers, and the total number ofwalkers Nw reduced by two. By keeping sorted listsof walkers, it is possible to do the search for possi-ble annihilation events using binary searches, withthe result that the annihilation step can be donewith O [Ns ln(NsNw)] effort.

At the end of the annihilation step, the lists ofsurviving newly-spawned walkers and parents aremerged. The merged list remains sorted, andbecomes the main list of walkers for the nexttimestep[28]. The annihilation can achieved in amemory efficient manner, keeping only one copy ofthe main list of walkers. In a one-electron basiswith 2M orbitals, [29], in discrete spaces (with a fi-nite spread of eigenvalues), repeated application ofeach walker needs d2M/32e + 1 four-byte integersof storage [30] to encode the occupation numberinformation of the determinant it lives on, as wellas the sign of the walker. For a simulation withNw walkers, therefore, the main walker list requires4Nw(d2M/32e+ 1) bytes of RAM. As will be seenin the applications, the number of walkers requiredto achieve convergence is typically smaller than thesize of the FCI space, often by a significant factor.Therefore the memory requirements of this algo-rithm are less severe than that of an conventionalFCI calculation.

It should be noted that with the above algorithm all(symmetry-allowed) determinants in the FCI space areaccessible, since no restrictions are placed on the spawn-ing step as to which determinants can be generated:given a walker on some determinant, any determinantconnected to it can be chosen to be spawned upon. Itfollows that the entire space of (symmetry-related) de-terminants can eventually be reached starting from anyone determinant. The fact that the entire space is ac-cessible enables the algorithm to converge onto the FCIwavefunction, which in general has non-zero amplitudeson all such determinants. Of course, it is also a straight-forward matter to impose a truncation in the CI space,for example, by excitation level, by simply not acceptingany spawning events at determinants beyond the speci-fied truncation level. In this way, one can perform calcu-lations equivalent to truncated CI, such as CISDTQ, etc.In large calculations, this approach can be used to pre-pare equilibrated or near-equilibrated ensembles of walk-ers at truncated excitation levels, before attempting asimulation in the full CI space.

5

A second point to note is that in the current algorithmeach walker attempts to spawn only once per timestep.It is possible to construct a modified algorithm in whicheach walker at each time step systematically attempts tospawn at all connected determinants (but with a differentspawning probability, given by ps(j|iα) = δτ |Kiαj|). Ob-viously, with this modification, the computational costper time step is much greater, although it might be ex-pected that a faster convergence is achieved. In tests,however, this version of the spawning algorithm provedto be less efficient overall than the one we outlined above.

A further point to note concerns time-step error. Theabove algorithm results in changes in populations on eachdeterminant proportional to δτ . The underlying propa-gator is G = 1− δτ(H − S). As has been pointed out in[29], in discrete spaces (with a finite spread of eigenval-ues), repeated application of such propagators startingfrom any initial state converges onto the exact ground-state as long as δτ ≤ 2/(Emax − S), where Emax is thelargest eigenvalue of H and S ≈ E0. Longer timestepslead to propagators in which excited-state contributionsdo not decay with time. In a non-stochastic implementa-tion of this propagator, convergence to the ground-stateis therefore guaranteed as long as the time-step does notexceeed this upper bound value. It is possible to estimateEmax as the energy of the most highly excited determi-nant available in a given one-electron basis, with Emax

increasing with the size of the basis.In practice, for the present stochastic algorithm, we

find that there is an additional constraint on the time-step which makes it to be smaller than the above value,namely that is it desireable that ps should not greatlyexceed unity, since this results in large walker blooms(i.e. multiple copies of walkers being suddenly spawnedon a determinant), which can lead to inefficient sampling.On the other hand, a small timestep leads to slow evo-lution and requires proportionately longer simulations,which is also inefficient. In practice, for the systems un-der consideration in this study, we find that that valuesfor δτ in the range 10−4 − 10−3 a.u. (with the smallervalues being used for larger basis sets), provides a rea-sonable balance between these two sources of inefficiency,and that we have explicitly verified that the correlationenergies we compute are insensitive of the value of thetime-step as long as they are in this range. In the re-maining sections we discuss some additional features ofour simulation methodology.

Constant S and Constant Nw simulations

If the shift S is kept constant at a value above Ecorr,then in general the number of walkers increases exponen-tially. It is therefore desirable to be able to do “constant”Nw simulation. This can be achieved by periodically (ev-ery A steps) adjusting the shift S according to the fol-

lowing simple prescription

S(τ) = S(τ −Aδτ)− ζ

Aδτln

Nw(τ)Nw(τ −Aδτ)

(17)

where Nw(τ) is the total number of walkers on timestepτ , and ζ is a damping parameter, i.e. if Nw has grownover the current update cycle, the shift is reduced (mademore negative), whilst if Nw has decreased, the shift isincreased (made more positive). Thus in “constant” Nw

mode, the shift S varies to attempt to keep walker num-ber constant. The damping coefficient ζ, prevents un-desirable large fluctuations in S. In constant S modethe number of walkers varies, typically increasing if S >Ecorr. If the distribution of walkers is correct (i.e. dis-tributed according to the ground-state amplitude), thenthe value of S which maintains the number of walkers isequal to the correlation energy of the problem. Con-versely, in constant Nw mode, the shift will fluctuateabout the correct correlation energy when the distribu-tion of walkers is correct. In the examples we reportbelow, we chose A = 5− 10, and ζ = 0.05− 0.1.

Simulation procedure and the projected energy

The simulations are started by placing a single walkeron the HF determinant. The simulation is then run for astart-up period, in which the shift S is kept at a constantvalue (of zero or small positive number). This resultsin the number of walkers initially growing very rapidly.When a desired number of walkers is reached, we startto adjust the shift S, as given by Eq. (17). It is ob-served that the value of the shift initially falls, and thenstabilises around a value about which it oscillates. Ifthe number of walkers is above a critical number (to bediscussed later), this value is the correlation energy ofthe system. It is desirable to have another measure ofthe correlation energy. This is provided by the followingprojection:

E(τ) =〈D0|He−τH |D0〉〈D0|e−τH |D0〉 (18)

= EHF +∑

j 6=0

〈Dj|H|D0〉Cj(τ)C0(τ)

(19)

= EHF +∑

j 6=0

〈Dj|H|D0〉Nj(τ)N0(τ)

(20)

where Nj(τ) is the number of walkers (defined by Eq.(13)) on Dj at timestep τ , and N0 is this number on D0

(which we assume to be the Hartree-Fock determinant).In addition, only the singles and double excitations areconnected to D0 (the singles being connected only in caseof open-shell systems), it is only the populations at theseexcitations that need to be evaluated for the purposes of

6

computing the energy, i.e.:

E(τ) = EHF +∑

j∈{Singles, Doubles}〈Dj|H|D0〉Nj(τ)

N0(τ)(21)

In the long-time limit, it is evident that E(τ) convergesonto the exact ground-state energy:

E0 = limτ→∞

E(τ) (22)

as long as the ratio Nj/N0 equals Cj/C0 of the FCI wave-function.

The two measures of the correlation energy, i.e. S andE(τ), are to a large extent independent of each other,since S depends on the total number of walkers, whereasE(τ) depends only on the population of walkers on theHartree-Fock determinant and the singles and double ex-citations, which typically consititute only a small fractionof the total walker population. In the limit of large walkerpopulations, the variations in S and E(τ) become largelyuncorrelated.

AN ILLUSTRATIVE EXAMPLE

Let us consider stretched N2, which is a strongly mul-ticonfigurational system. For the purposes of illustrationof how the exact wavefunction emerges during the courseof the simulation, we consider here a very small basis,with 379 Slater determinants, which allows a convenientvisualisation of the evolution of the walker distributionas the simulation proceeds. The behaviour of the shift,and of the projected energy, are shown in Fig. 1. The in-stantaneous energy for each timestep (Eq. 20) convergeson the FCI energy within 10,000 steps. In this simula-tion, we started varying the shift at step 10,000, and theshift very rapidly decreases from zero towards the FCIenergy, and stabilises at the correct value. The two in-dependent measures of the correlation energy agree tobetter than 1 mH. In real applications, where the exactFCI energy is not known, the co-incidence of the thesetwo measures provides a strong degree of confidence inthe obtained correlation energies. Equally remarkable isthe convergence of the walker distribution onto the exactwavefunction: An animation of the time evolution of thissimulation (as well as other systems) is also provided as amovie held on our website [31]. In the third panel of Fig.1 we also show the behaviour of the simulation if the an-nihilation step is removed. In this case, the shift does notstabilise at the correct value, and the instantaneous en-ergy fluctuates wildly. Without annihilation, the walkerdistribution cannot converge onto to correct ground-statedistribution; there is no “interaction” between the posi-tive and negative walkers, and in fact the distribution ofthe positive walkers among the determinants increasinglymatches the distribution of negative walkers as the sim-ulation proceeds. In other words, in common with the

classic sign problem, the signal to noise ratio decreasesexponentially quickly. In addition, because the shift goesto the wrong (and much too negative) value, the rate ofdeath processes becomes very large, leading to indiscrim-inate death of walkers, essentially irrespective of whichdeterminant they occupy. The walker distribution is notable to build a signal on the determinants with signifi-cant weight. This example shows the crucial role playedby the annihilation step. In fact the annihilation stepallows a symmetry-breaking to occur, in which one ofthe two allowed solutions, either +Ψ or −Ψ, is settledupon. Without providing any information regarding thesign structure of the FCI wavefunction, our algorithm iscapable of settling down onto the correct wavefunction.In a real sense, the many-body wavefunction has emergedduring the course of the simulation. The breaking of the±Ψ symmetry is a necessary condition for this emergence.An analogy can be made with phase-transitions in classi-cal spin-systems. The ordered phase (e.g. ferromagneticor antiferromagnetic) of a spin-system is is degeneratewith respect to overall sign-change of all spins. For thesystem to acquire net magnetisation (or staggered ma-gentisation) below the critical point, the symmetry mustbe spontaneously broken, and the ordered phase, whichhas a symmetry lower than that present in the Hamilto-nian can then emerge.

This example demonstrates that the method in prin-ciple works exactly. The question now is: it is capableof working for much larger FCI spaces? To answer thisquestion, we have studied a number of molecular systems(Ne, H2O, C2, N2. . . ) in a variety of basis sets, to whichwe now turn.

RESULTS

We next consider a range of molecules, treated inmore realistic Dunning basis sets[32]. We also performa benchmark study of the all-electron Ne atom in sev-eral families of basis sets, namely cc-pVXZ, aug-cc-pVXZand cc-pCVXZ (X=D,T,Q). The size of spaces consideredrange from ∼ 106 to ∼ 1014.

In spaces of such size, the pattern of walker growthgenerally changes. Keeping the shift fixed, initially thewalker number grows exponentially. However, at a sys-tem specific value, this exponential growth attenuates,apparently suddenly, and the number of walkers hits aplateau. After a period of essentially zero growth in over-all walker number, the growth picks up again, after whichpoint we start adjusting the shift. An example of this isshown for the all-electron water molecule treated with acc-pVDZ basis in Fig. 2.

During this plateau phase, the rate of walker birth isexactly matched by the combined rate of walker deathand walker annihilation. Before reaching the plateau, therate of annihilation is small and therefore the overall rate

7

0 50 100 150 200 250 300 350 400�0.4

�0.20.00.20.40.6

With AnnihilationExact ��

(�=50,000)

0 10000 20000 30000 40000 50000�2.0

�1.5

�1.0

�0.50.00.5

ShiftE(�)EFCI

0 10000 20000 30000 40000 50000Iterations

�6�5�4�3�2�101 Without Annihilation

ShiftE(�)EFCI

FIG. 1: Comparison of sampling of determinants compared to full diagonalisation result for a stretched N2 molecule in a spaceof 379 determinants. Ψ indicates the normalised walker number on each of the determinants after 50,000 iterations, comparingit to the exact wavefunction. The shift and E0 values for each iteration are shown in the lower plot.

of walker growth is positive. At the plateau, sufficientnumbers of walkers begin to annihilate each other, andthis leads to the attenuation of walker number growth.Very significantly, owing to walker annihilation, we ob-serve that during the plateau phase, the sign structure ofthe FCI wavefunction is converged upon, in every casewe have studied. As long as we have a larger numberof walkers in the system than the number defined by theplateau position, Nc, then when we allow the shift to vary(in constant Nw mode), it converges onto the correlationenergy. This Nc value is therefore an important, system-dependent parameter which indicates the required sam-pling of the space. If this sampling can be achieved, thenthe correct correlation energy for the system can be con-fidently converged upon to the desired accuracy. This Nc

value in all cases is smaller than the size of the completespace, often substantially so. We can therefore describe aparameter fc which gives us the relative number of walk-ers required, compared to the number of determinants inthe full space (fc = Nc

NFCI). fc is one measure of the dif-

ficulty the method has in achieving convergence. Wherefc is small (or even zero), convergence can be achieved

with a relatively small number of walkers.It turns out that the value of Nc (for a given system)

is remarkably insensitive to the initial conditions, andmethod of equilibration, of the walkers. Thus, we havefound that we need the same number of walkers to attainthe plateau if we start the simulation with one walkerand let the number grow rapidly throughout the entireFCI space, or if we gradually enlarge the space, e.g. viaexcitation level. This indicates that Nc is an intrinsicparameter which characterises the system, for a givenone-electron basis.

In Fig. 3 we show the convergence of the averaged pro-jected energy 〈E(τ)〉 as the simulation proceeds, in thecase of the water molecule. Also included, is the abso-lute error |〈E(τ)〉 − EFCI| from the known FCI energy.It can be seen that there is an exponential convergenceto an accuracy of under 0.1mH of the energy. To givean idea of the computational cost of the method, theCPU time taken to converge all-electron cc-pVDZ wa-ter to this accuracy with our current code is three hoursusing a quad-core (2.66GHz) PC.

Apart from the water molecule, we have tested our al-

8

0e+00

1e+07

2e+07

3e+07

4e+07

Num

ber

of

walk

ers

Total walker numberCreated walkersDied walkersAnnihilated walkers

0 5000 10000 15000 20000 25000 30000Iterations

�0.2

�0.1

0.0

0.1Energ

y/H

art

rees

ShiftEcorr

FIG. 2: A typical trend of walker growth for H2O in a cc-pVDZ basis set in constant shift mode (S = 0) until iteration16,970 , showing the appearance of a well-defined plateau in Nw, in this case at Nc = 26 × 106 walkers. For this system,NFCI = 451× 106. The number of walkers at the plateau is therefore about 6% of the FCI space. Also shown on the plot arethe number of walkers created, died and annihilated per A = 10 iterations. It is evident that at the plateau, the combinedrate of death and annihilation matches the birth rate from the spawning. The plateau gradually gives way to a growth phasein walker number, which increases exponentially. Once in the growth phase, the shift is then allowed to vary according to Eq.(17), (in this example on iteration number 16,970). The number of walkers then rapidly stabilises. The lower plot shows thatthe value for the shift which stabilises the walker growth is exactly the correlation energy for the system.

�0.20

�0.15

�0.10

�0.05

0.00

Energ

y /

Hart

rees

�E(�)�EFCI

0 2000 4000 6000 8000Iterations

10-6

10-5

10-4

10-3

10-2

10-1

Energ

y /

Hart

rees

|�E(�)��EFCI|

FIG. 3: Convergence of the energy estimator averaged over all previous iterations, showing an exponential convergence to theEFCI for a cc-pVDZ water system at equilibrium geometry. The lower plot shows the absolute difference between the energyand the exact energy on a logarithmic scale. δτ = 1× 10−3 a.u. In this example, the simulation was started by placing a singlewalker at the Hartree-Fock determinant and allowing the population to grow throughout the entire space until the post-plateaugrowth phase has been reached.

9

gorithm in other systems for which FCI results are avail-able for comparison, shown in Table I, namely Ne atom,C2 and N2 at equilibrium and at stretched geometry. Ascan be seen, the computed energies are to within 1 mHof the published FCI results for all systems considered,confirming the ability of the method to reproduce theFCI energies across a broad range of systems.

There is considerable variation in fc among these sys-tems. For the Ne atom, and all-electron H2O, fc ¿ 0.1,whereas for C2 and N2, fc ∼ 0.5, i.e. comparable thoughstill smaller than the FCI space. The behaviour of thetwo N2 systems is particularly instructive. As the N2

bond is stretched from its equilibrium geometry, the de-scription of the problem changes from one which is es-sentially single-reference, to one which is highly multi-reference with significant contributions up to hextupleexcitations. This makes an accurate description of thebinding curve of nitrogen particularly challenging formany methods. Surprisingly, the results from these twosystems show a similar fc, indicating that the ease withwhich the method achieves convergence is not simply re-lated to the dominance of the Hartree-Fock determinant,and that the number of significant determinants is not thekey factor in its efficiency. This feature can hopefully beexploited in the study of multiconfigurational systems.The fact that in all cases fc is less than one, sometimessubstantially so, means that one needs fewer walkers thandeterminants to achieve converge the calculation. . Atany one time step, only a fraction of the space is occu-pied, but as long as fc has been exceeded, averaged overtime each determinant is correctly sampled according toits amplitude. We have exploited this property to studysystems for which the FCI space is extremely large.

In Table II, we report the energies of several molecules,including CO, CH4, O2 and NaH – for which we have notbeen able to find published FCI energies, and thereforeserve as predictions which could be verified by explicitFCI calculations. Included in this list are the open-shellsystems CN(2Σ+) and O2(3Σ−g ) . As can be seen fromthe table, the size of FCI spaces for these systems rangefrom a few million to over 1011. Particularly striking arethe variations in fc. The worst case is for CH4, wherewe find an fc = 0.898, whilst for NaH fc = 3 × 10−4.Remarkably, in this case, only 64 million walkers wereneeded to converge the energy for a space of 205 × 109

determinants. Results from coupled-cluster with pertur-bative triple excitations, CCSD(T), have been includedfor comparison [38], and as can be seen our correlationenergies are generally slightly larger (with the notableexception of CH4), although since CCSD(T) is not vari-ational, not too much can be read into this result. Wedo note that the agreement between the two is generallybetter for closed-shell systems than open-shell ones.

An important question with the present method ishow the plateau varies for a given system as the basisset is improved, since we typically want to do calcula-

tions in substantial basis sets, which improve accuracyand allow extrapolation to the complete basis-set limit[39]. In Table III, we present a study of the Ne atomin various basis-sets, to investigate the fraction fc re-quired to achieve the plateau. Several trends are dis-cernible. Within a given type of basis set, the value offc is fairly constant, but there are marked variations be-tween different families. Thus cc-pVXZ (denoted VXZ inTable III, where X=D,T,Q) family have fc ≈ 5 × 10−4,whereas the aug-cc-pVXZ (AVXZ) family have a largerfc ≈ 1.5 × 10−3, whilst the cc-pCVXZ (CVXZ) havea very favourable fc ≈ 2 × 10−5. The latter are alsogive the most accurate energies. Clearly inclusion of coreelectrons with CVXZ basis sets is very favourable for thepresent method. Extrapolation to the complete basis-setlimit [39] yields a correlation energy of 393.0 mH, which iswithin 2.5 mH of the “exact” non-relativistic result [40].Such extrapolated energies are of course not variational,and are known to yield errors of about 5 mH, consistentwith what is observed here.

These results illustrate that the memory requirementsof the method are substantially lower than a conventionalFCI calculation. For example, the all-electron Ne atomin a cc-CVQZ basis required Nc = 2.2 × 109 walkers,which amounts to 62 Gbytes of RAM. By contrast, a FCIcalculation of the same system would require a minimumof two vectors of length NFCI = 131 × 1012, i.e. over2× 106 Gbytes, a vastly greater amount of storage.

Since in general Nc scales linearly with NFCI , thisimplies that the present method has exponential scalingwith N and M . The method, therefore, does not consti-tute a “solution to the sign problem”, for which a poly-nomial scaling is required. It is perhaps best thought ofas an alternative method to FCI, with a smaller prefactor(proportional to fc), which in some cases is substantiallyso. It remains to be seen if variations on the present al-gorithm can be found in which either the prefactor canbe further supressed, or even better, if the scaling can beimproved upon.

CONCLUSIONS

We have described a new Quantum Monte Carlo algo-rithm in Slater determinant space, which we show is ableto converge on the FCI energy of the system under con-sideration without any a priori information, as long as asystem-dependent critical number of walkers is exceeded.The exact FCI wavefunctions emerge spontaneously oncethis critical number has been reached. Walker annihila-tion is shown to play a key role in this process. Thecomputational requirements of the method are largelydetermined by the value of this critical number. If it canbe reduced for a given system, e.g. by a judicious choiceof orbitals obtained via orthonormal transformations ofthe canonical Hartree-Fock orbitals, then the computa-

10

System (N,M) NFCI/106 Nc/106 fc Etotal EFCI Reference

Ne: aug-cc-pVDZ (8,22) 6.69 0.21 0.031 -128.70949 -128.709476 [33]

C2: cc-pVDZ (8,26) 27.9 15.0 0.538 -75.7299 -75.729853 [34]

H2O: cc-pVDZ (10,24) 451 26 0.058 -76.24186 -76.241860 [35]

N2-eqm: cc-pVDZ (10,26) 541 270 0.499 -109.27649 -109.276527 [33]

N2-stretched: cc-pVDZ (10,26) 541 345 0.637 -108.9669 -108.96695 [36]

TABLE I: Results for systems with FCI comparisons. The geometries for the N2 molecule were eqm: 2.068a0, stretched: 4.2a0,and C2: 1.27273A. The geometry for the water molecule was taken from [35]. The working space includes all point groupsymmetry of the molecule from D2h or the largest available subset thereof. All systems had core electrons frozen apart fromH2O. NFCI is the size of the FCI space in the D2h point group (C2v for H2O). The digit in italics for Etotal, represents the firstuncertain digit. Nc is the number of walkers required to achieve the plateau. fc = Nc/NFCI.

System (N,M) NFCI/106 Nc/106 fc Etotal ECCSD(T)

Be: cc-V5Z (4,91) 2.11 0 0 -14.64638(2) -14.64629

CN: cc-pVDZ (9,26) 246 173 0.704 -92.4938(3) -92.49164

HF: cc-pCVDZ (10,23) 283 0.998 0.0035 -100.27098(3) -100.27044

CH4: cc-pVDZ (8,33) 419 377 0.898 -40.38752(1) -40.38974

CO: cc-pVDZ (10,26) 1,080 777 0.719 -113.05644(4) -113.05497

H2O: cc-pCVDZ (10,28) 2,410 47.4 0.0196 -76.28091(3) -76.28028

O2: cc-pVDZ (12,26) 5,409 2,651 0.490 -149.9875(2) -149.98562

NaH: cc-pCVDZ (12,32) 205,300 63.8 0.00031 -162.6090(1) -162.60901

TABLE II: Predicted FCI results. The geometries of the molecules were (in Angstroms): CN(1.1941), HF(0.91622),CH4(rCH=1.087728), CO(1.1448), H2O(rOH=0.975512, θ=110.565) [35], O2(1.2074) and NaH(1.885977). CN and O2 orbitalswere constructed from a restricted open-shell HF calculation with a spin-multiplicity of two and three respectively. CN, CH4,CO and O2 had frozen core electrons. The number in brackets represents the error in the previous digit, obtained through aFlyvbjerg-Petersen blocking analysis [37] of E(τ).

.

tional effort required to compute the correlation energywill be similarly reduced. We are currently investigatingsuch ideas.

Once this critical number of walkers has been reached,the correlation energy can be computed with confidence.Agreement between two estimators of the correlationenergy provides additional support for the values ob-tained. We have reproduced existing FCI energies, aswell as reporting new systems with very large FCI spaces.Favourable memory requirements, as well as ease of par-allelisation, are attributes of this algorithm which shouldenable yet larger systems to be tackled in the near future.

In common with the FCI method, the only systematicerror in the calculation of the correlation energy arisesthrough basis-set incompleteness, which can however besystematically improved via explicit calculation on largerbasis sets, together with extrapolation to the completebasis-set limit. This is marked contrast with the uncon-trolled approximation of fixed-node QMC where the errorintroduced is hard to reduce systematically.

The present method provides a synthesis of QuantumMonte Carlo and quantum chemistry. Single and multi-reference problems can both be tackled, and the difficultyof the procedure does not seem to be closely tied to thisclassification of systems. This suggests that future ef-fort will be primarily focused on multi-reference systems

which provide a sterner challenge to many other methods.In addition to this, several extensions of the present tech-nique can be envisaged which should allow much largersystems to be treated. These include CASSCF method-ologies, perturbation theory extensions, and the use ofconfiguration state functions, which are all currently un-der investigation.

APPENDIX A

Two determinants |Di〉 and |Dj〉 are said to be coupledif and only if 〈Di|H|Dj〉 6= 0. In our applications, His a molecular Hamiltonian with one-electron and two-electron terms:

H =N∑

i

hi +∑

i<j

1rij

, (23)

hi = −12∇2

i + vext(ri) (24)

Determinants which differ by three or more (orthonor-mal) spin-orbitals are therefore uncoupled, and we needonly consider pairs of determinants which differ by twoor fewer spin-orbitals, i.e. determinants which are doubleor single excitations of each other. Such matrix elements

11

Basis Set Orbitals NFCI/106 Nc/106 fc/10−3 Ecorr

VDZ 14 0.502 0 0 0.19211(4)

CVDZ 18 9.19 0 0 0.23365(3)

AVDZ 23 142 0.248 1.7 0.21510(3)

VTZ 30 2540 0.506 0.199 0.28341(9)

CVTZ 43 116,000 2.3 0.0198 0.33628(2)

AVTZ 46 235,000 338 1.43 0.2925(4)

VQZ 55 1.51×106 681 0.451 0.3347(10)

CVQZ 84 119×106 2200 0.0185 0.3691(1)

Extrap. 0.3930

Exact 0.3905

TABLE III: Ne atom results in Dunning[32] basis sets, giving correlation energies in Hartrees. All electrons (10) were correlatedover all excitation levels. The “exact” result is the non-relativistic infinite-nuclear mass corrected experimental value, calculatedin reference [40]. The extrapolated result for comparison is obtained using the technique by Halkier et al. [39] using the cc-pCVTZ and cc-pCVQZ correlation energies.

can be computed using the Slater-Condon rules [41–43]as follows. In the case of double excitations, let Dj bethe following double excitation of Di:

|Dj〉 = a†ra†saqap|Di〉 (25)

Then

〈Di|H|Dj〉 = 〈rs||pq〉 ≡ 〈rs|pq〉 − 〈rs|qp〉 (26)

where the (2-electron) 4-index integrals are defined by

〈rs|pq〉 =∫

dr1dr2φ∗r(r1)φ∗s(r2)

1r12

φp(r1)φq(r2) (27)

Similarly, in the case of single excitations, let |Dj〉 =a†rap|Di〉, then

〈Di|H|Dj〉 = 〈r|h|q〉+′∑

k

〈rk||qk〉 (28)

where the sum over k extends over the N−1 spin-orbitalscommon to Di and Dj.

The diagonal matrix elements are given by:

〈Di|H|Di〉 =∑

p∈i

〈p|h|p〉+12

∑

p,q∈i

〈pq||pq〉 (29)

In the present work, all the necessary integrals (4-indexand 2-index) were generated from restricted Hartree-Fockorbitals using a modified version of QChem [38].

APPENDIX B

In our algorithm, it is necessary to be able to gener-ate all single or double excitations of any determinantin such a way that its generation probability, pgen(j|i) iscomputable, non-zero and normalised. We employed thefollowing simple strategy (which is probably not optimal

from a sampling perspective; but this consideration is leftfor a future study).

Let us consider the generation of |Dj〉 = a†ra†saqap|Di〉,

which involves the selection of the occupied pair (p, q)and the unoccupied pair (r, s) (with respect to Di). Then

pgen(j|i) = pgen(r, s|p, q)pgen(p, q) (30)

where pgen(p, q) is the probability to select the orbitalpair (p, q) in Di, and pgen(r, s|p, q) is the probability toselect the orbitals pair (r, s) given that we have selected(p, q). For an N electron system, we select the occupiedpair (p, q) with uniform probability, i.e.:

pgen(p, q) =(

N

2

)−1

= 2[N(N − 1)]−1 (31)

We further write:

pgen(r, s|p, q) = pgen(r|s, p, q).pgen(s|p, q)+pgen(s|r, p, q).pgen(r|p, q) (32)

In other words, we select s with probability pgen(s|p, q),and r with probability pgen(r|s, p, q), before computingthe probability that we could have picked the unoccu-pied orbital r first, pgen(r|p, q), followed by s, to ob-tain the same excitation. Generally, pgen(r|s, p, q) 6=pgen(s|r, p, q). The advantage of this approach is thatwe can combine spin and symmetry information, whichis usually available in the form of the irreducible repre-sentation spanned by each spatial orbital. For example,in the generation of r, given (s, p, q), the irreducible rep-resentation (Γr) of r is dictated to be the direct productof the irreps of (s, p, q):

Γr = Γs ⊗ Γp ⊗ Γq (33)

[In the case of Abelian groups this uniquely determinesΓr. In this study of atoms and homonuclear diatomics,

12

we used D2h as such a group]. Having selected s, wethen determine the irreducible representation of r, andthen explicitly count how many unoccupied orbitals ofthis irreducible representation are available for selection(imposing, in addition, that the net change in spin-polarisation is zero). The reciprocal of this number givespgen(r|s, p, q). This completes the generation algorithmfor double excitations.

The same method can be used to compute the genera-tion probability of a single excitation, though it is muchsimpler. Given |Dj〉 = a†rap|Di〉, we need:

pgen(r, p) = pgen(r|p).pgen(p) (34)

We select p uniformly out the occupied orbitals of Di,(i.e. pgen(p) = N−1) and then select r such that Γr =Γp. We then count the number of unoccupied orbitalsof this irreducible representation that are available withthe same spin, and compute pgen(r|p) as the reciprocalof this number. The probability of choosing to createa double excitation is given by Pd, where 0 < Pd < 1and hence the probability of choosing to generate a singleexcitation is 1−Pd. Pd is chosen to approximately reflectthe relative number of double excitations compared tosingle excitations. To obtain the final pgen(j|i), we needto multiply by Pd if it is a double excitation, or 1−Pd ifit is a single excitation to maintain normalization of theprobabilities.

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